Development of improved estimators of finite population mean in simple random sampling with dual auxiliaries and its application to real world problems

In general, the incorporation of supplementary information reduces the Mean Square Error (MSE) and, consequently, enhances the precision of estimating a population parameter. This improvement relies on the appropriate application of a suitable function, with careful consideration. This study introduces two innovative families of estimators for the finite population mean, both of which exhibit superior performance in scenarios involving dual auxiliary information in simple random sampling. Expressions up to the first-order approximation, for bias, and Mean Square Error were derived, and the conditions under which these proposed families surpassed the existing estimators. Our evaluation involved the use of both real and simulated data to compute the Mean Square Error and Percent Relative Efficiency (PRE) of the estimators. A comparative analysis revealed that under the specified conditions, both proposed families yielded more precise results.


Introduction
Incorporating supplementary data alongside the primary study variable typically enhances the efficiency of the estimation techniques.Generally, the efficiency tends to improve as the number of auxiliary variables increases.The utilization of auxiliary information can significantly bolster the precision of population parameter estimators provided that an appropriate mathematical function is diligently applied.When estimating the finite population mean, various methods such as ratio, regression, and exponential estimators come into play when there is a direct correlation between the study variables and auxiliary variables.Conversely, a product estimator is employed when the correlation is negative.
Building upon the foundation work of [17,25] this study introduces two innovative families of exponential ratio-cum-product estimators.The objectives of this study are twofold.
Introducing new, precise families of population mean estimators that encompass both ratio and product estimators.
To incorporate additional population parameters such as correlation coefficients and coefficients of variation from auxiliary variables to further enhance the precision of the proposed estimator.
Hence, this study delves into the use of dual auxiliary data and evaluates the estimator's effectiveness.The remainder of the manuscript is structured as follows: Section 2 contains the methodology; existing estimators and their MSEs are discussed in Section 3; Section 4 presents the suggested estimators along with the development of bias and MSE Expressions, Section 5 details the conditions under which the proposed families outperform existing ones; and Sections 6 and 7 encompass empirical and simulation studies, respectively.Section 8 presents a discussion and conclusions.The limitations are presented in Section 9, respectively.

Methodology
Let a finite population denoted as Ω = (ω 1 , ω 2 , ω 3 , ....., ω N ) with a total size of N. We aim to obtain a random sample of n units using the Simple Random Sampling (a sampling scheme where each unit of the population has an equal probability to enter the sample) without replacement (SRSWOR) method.Within this context, we have a study variable denoted as "y" and our objective is to estimate its mean.To assist in this estimation, we have access to information from two auxiliary variables, "X" and "Z" It is important to note that data pertaining to both the main study variable and the variables are readily accessible.
can satisfy the following properties.
The integration of auxiliary information not only contributes to reducing bias but also minimizes variability in the estimation, ultimately leading to improved overall performance.Therefore, the dual auxiliary method serves as a robust means to enhance estimator efficiency by harnessing the power of additional information in the estimation process.

Estimators in literature
Numerous estimators for calculating the population mean have been developed and documented in the field of statistics.Researchers and practitioners often choose from this pool of estimators based on the specific characteristics of their data and the underlying assumptions that align with their research objectives, ensuring the most suitable and accurate estimation of the population mean.
The customary, traditional estimate of mean EST0 = y is used when the data consists only a study variable.The estimator's variance is provided in Eq. (1) as According to [29], the ratio estimator in the situation of a dual auxiliary is given in Eq. ( 2) as follows: The MSE of the above ratio estimator is listed in Eq. (3) as The chain ratio-product estimator proposed by [35] is given in Eq. ( 4) as MSE up to first order of approximation of the above estimator is shown in Eq. (5) as The regression estimator in dual auxiliary is given in Eq. ( 6) as Where b 1 = The MSE of the regression estimator takes the following form of Eq. ( 7) [21] proposed the following regression-exponential-Ratio type estimator mentioned as in Eq. ( 8) Where w 1 and w 2 are minimizing constants and α takes values either 1 or 0 to have ratio exponential or product exponential estimator respectively.The expression for the least MSE of the estimator is provided by Eq. (9). Where The proposed estimator by [12] in two auxiliary takes the exponential form listed in Eq. ( 10) as follow Where w 3 , w 4 and w 5 are the minimizing constants the values of whom are to be obtained so that the resulting MSE is minimum.where u and v are generalizing constants, whose values consists of different parameters of the auxiliary variables.Let η = uX uX+v then minimum MSE of estimator is listed Eq.(11) as Where [30] Proposed the following product estimator in Eq. ( 12) Here the w 6 , w 7 and w 8 are the minimizing constants values of whom are obtained so that the MSE is least, α 1 and α 2 can take any value either negative or positive.For w 6 = Ψ0 Ψ , w 7 = Ψ 1 Ψ , and w 8 = Ψ2 Ψ the minimum MSE of the Singh and Nigam's estimator is provided in Eq. (13) as Where

Proposed estimator
This segment Presents two families of suggested estimators for the finite population mean through the utilization of double secondary variables within the framework of simple random sampling.Furthermore, within this segment, we derive expressions for both bias and MSE.[25] proposed exponential ratio-cum-product estimator in single auxiliary case which gives efficient results even if the correlation between the variables is not strong enough.

First proposed estimator
Here φ 1 and φ 2 are constants whose values are to be determined so that the resultant MSE is minimum, u and v are generalizing constants which can adopt any value from the known values of the parameters of the supplementary variable.[21] Introduced the below estimator in the presence of two auxiliary variables.
Again in this estimator the φ 3 and φ 4 are minimizing constants whose values are obtained by the differentiating the MSE expression and α can take value either 0 or 1.
Motivated from these works a novel family of estimators for population mean is suggested in Eq. ( 14) below by modifying [21].
Several estimators can be produced from the above estimator by substituting different values of u, v and l .here t 1 and t 2 are the minimizing constants whose values are determined by minimizing the MSE and u, v and l are the generalizing constants which can assume any suitable value or any known parameter of the population.For obtaining the MSE of the estimator, we can express the above Eq.( 14) in terms of sampling errors as below Simplifying Eq. ( 15) by applying Taylor series the following equation is obtained in terms of sampling errors Where Now the expression mentioned below is obtained by subtracting Y from each side of Eq. ( 16) After taking expectation on Eq. ( 17) the following bias expression is obtained as in Eq. ( 18) Let's take square and expectation on Eq. ( 17) to have an expression for MSE Or the above equation can be written in Eq. (19) as The simplified form of the above equation is provided in Eq. ( 20) as follow Where Applying mathematical rule to obtain the optimum values of t 1 and t 2 from Eq. ( 20) as follow Solving Eq. ( 21) and Eq. ( 22) simultaneously the following values of t 1 and t 2 are obtained..withthese values the minimum MSE of the estimator adopts the following form This MSE Eq. ( 23) is supposed to be minimum relative to the other competing estimators for finite population mean for different values of u, v and ℓ.

Second proposed estimator
A ratio cum product estimator was introduced by [4] for the finite population mean in the existence of one supplementary variable.
Modifying by [4] incorporating the idea of [25] and using the dual auxiliaries a novel and effective modified exponential estimator is produced as.
Where in Eq. ( 24) t 3 and t 4 are the minimizing constants and ℓ, u and v are the same as defined in section 4.1.The following different forms of estimators are obtained by putting different values of the l , u and v.
Table 1 shows some of the all estimators that can be deduced from the proposed families of the estimators.To find the MSE of the ESTp2 estimator we express Eq. ( 24) in terms of sampling errors as follows

Table 1
Different estimators deduced from ESTp1 and ESTp2.

S. No
] exp ] exp After simplifying Eq. ( 25) the difference equation is given by Where The Bias of the estimator up to first order of approximation by taking the expectation of Eq. ( 26) is given in Eq. (27) as Taking square of Eq. ( 26) and applying expectation we have Applying mathematical rules on Eq. ( 28) to find the optimum values of t 3 and t 4 as follow and Solving both the equations Eq. ( 29) and Eq. ( 30) simultaneously, we have Where ) and Therefore, after putting these values the smallest possible MSE is given as This Eq. ( 31) is the second proposed family of estimators for the mean of a finite population under simple random sampling with two auxiliary variables that is deemed efficient under the given conditions for various values of u, v and ℓ.

Theoretical comparison
The efficiency of the suggested estimator could be judged through the following conditions in relation to the existing estimators.
K. Sher et al.

For second proposed estimator
The second estimator we introduce in this article is anticipated to exhibit efficiency superior to that of all other estimators, provided the specified conditions are met.

Numerical comparison
To evaluate the performance of the suggested estimator on numerical data the below five real world data sets have been considered.Table 3 shows the MSEs of the existing and proposed estimators for all five datasets given in Table 2. Table 4 presents the PREs of the estimators obtained with respect to the usual estimator of the mean.It is obvious from both tables that in all the above five datasets, the proposed estimators (ESTp1 and ESTp2) surpass all the other estimators; in particular, its efficiency is increased using the transformation.Using different values of the parameters of the auxiliary variables, the efficiency can be further increased.
Fig. 1 shows the PREs of the estimators relative to the classical estimator for the population mean, where each line shows the PRE values for different datasets.The graph shows that the PREs are higher for all six estimators (ESTP1(1) to ESTP2(3)) compared with the existing estimators in all datasets.Hence, the suggested families outperform all the other contenders in terms of efficiency when estimating the population mean.

Simulation study
This section aims at the simulation study of the estimators to check the stability of the estimator in haphazard from sample to sample.Population of size 1000 is taken from multivariate normal distribution for a study variable y and two auxiliary variables X and Z, with mean vector and covariance matrix as where MSE and PRE values for the existing and proposed estimators were carried out using following steps in R software.
Step-1.Simple random samples without replacement (SRSWOR) of different n = 10, 20, 50, 100, 200.have been drawn from the target population.For each sample size a loop of 10,000 times caried out and allowed R-studio to compute the values of the estimators at each iteration.
Step-2.For each sample, values of the existing and proposed estimators have been carried out separately by taking average of all the

iterations.
Step-3.Using values caried out in step-2 the MSE of the estimators are obtained.
Step-4.PREs of the estimators are obtained using the following formulae.
pre(EST i ) = Var(EST0) MSE(ESTi) × 100 where EST i replaces different estimators.Table 5 presents the MSEs of the estimators for all five sample sizes of the simulated data.Table 6 presents the simulation results of the different estimators in simple random sampling with two auxiliaries with respect to the usual estimators for various sample sizes.By exploring the tables, it can be observed that the MSEs of the suggested estimators (ESTp1 and ESTp2) are smaller and the PREs of the suggested families of estimators are higher relative to all the other estimators of the population mean.Further, our estimator is stable with respect to the sample; as the sample size increases, the estimator's efficiency also increases.Therefore, our proposed estimator stands out as the superior choice among the competing estimators under investigation.
Fig. 2 shows the PREs of the estimators relative to the classical estimator for each sample size for the simulated data.It is obvious from the graph that the PREs for the six proposed estimators (ESTP1(1) to ESTP2(3)) in the simulated data are much higher than those of the existing estimators.Hence, the proposed families are efficient at estimating the population mean among all competitors.

Discussion and conclusion
In this study, we introduce two distinct families of exponential-type estimators tailored for application in the context of simple random sampling with dual auxiliary variables.Our analysis extends to the first-order approximation, allowing us to derive expressions for key properties, such as bias and MSE.Notably, we identified the conditions under which our proposed families of estimators outperformed all existing competitors in terms of the MSE.
The effectiveness of these novel families of estimators was further confirmed through rigorous testing of real-world datasets.Our findings consistently revealed lower MSE and higher PRE values compared with all other estimators considered for the same parameter.This underscores the robustness and efficiency of the proposed estimators when applied to practical scenarios.
To ensure the reliability and stability of our estimators across various sample sizes, we conduct extensive simulation studies.Our results, stemming from ten thousand simulations drawn from a tri-variate normal population and spanning five distinct sample sizes (n = 10, n = 20, n = 50, n = 100, n = 200), underscored the efficiency and stability of our proposed estimators under the specified conditions.
In summary, our comprehensive examination of these estimators, supported by both real data analysis and simulation studies, confirms their superiority and efficiency relative to existing competitors in the field.These findings collectively underscore the practical utility of our proposed estimators in simple random sampling with dual auxiliary variables.

Limitations
Following are some limitations of the proposed estimators.
• These estimators are obtained for the finite populations there is no guarantee that these estimators will be efficient under infinite populations.• These estimators are designed for only two auxiliary variables scenarios.

syx s 2 x and b 2 = syz s 2 z
are the sample regression coefficient associated with β 1 =

Table 2
Summary statistics of data.

Table 3
MSE's of the estimators on real data.

Table 4
PRE's of the estimators relative to the classical estimator.
1.Plotting PREs against each estimator for all populations.K. Sher et al.

Table 5
MSEs of the estimators for simulation data under different sample sizes.

Table 6
PREs of the estimators with respect to usual estimator for simulated data under different sample sizes.
Fig. 2. Plotting PREs in different sample size for simulated data.K.Sher et al.